The nonnegative rank of a matrix: Hard problems, easy solutions

TL;DR

This paper introduces elementary linear algebra techniques to address two key problems in nonnegative matrix theory: proving NP-hardness of nonnegative rank and solving the open problem of rational factorizations.

Contribution

It provides a simple proof of NP-hardness for nonnegative rank and offers a solution to the longstanding open problem of rational nonnegative factorizations.

Findings

Nonnegative rank is NP-hard to compute.

Rational nonnegative factorizations can be explicitly constructed.

Elementary linear algebra suffices for these solutions.

Abstract

Using elementary linear algebra, we develop a technique that leads to solutions of two widely known problems on nonnegative matrices. First, we give a short proof of the result by Vavasis stating that the nonnegative rank of a matrix is NP-hard to compute. This proof is essentially contained in the paper by Jiang and Ravikumar, who discussed this topic in different terms fifteen years before the work of Vavasis. Secondly, we present a solution of the problem of Cohen and Rothblum on rational nonnegative factorizations, which was posed in 1993 and remained open.